University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
2
2018
06
01
Combinatorial parameters on bargraphs of permutations
1
16
EN
Toufik
Mansour
Department of Mathematics, University of Tennessee, Knoxville, TN, USA
toufik@math.haifa.ac.il
Mark
Shattuck
Mathematics Department, University of Tennessee, Knoxville, TN, USA
mark.shattuck@tdt.edu.vn
10.22108/toc.2017.102359.1483
In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.
combinatorial statistic,$q$-generalization,bargraph,permutations
http://toc.ui.ac.ir/article_22243.html
http://toc.ui.ac.ir/article_22243_ee9a92039072d73f603a278c71ef4387.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
2
2018
06
01
The log-convexity of the fubini numbers
17
23
EN
Qing
Zou
The University of Iowa
zou-qing@uiowa.edu
10.22108/toc.2017.104212.1496
Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence ${sqrt[n]{f_n}}_{nge 1}$ by using the aforementioned bounds.
Fubini number,log-convexity,monotonicity
http://toc.ui.ac.ir/article_21835.html
http://toc.ui.ac.ir/article_21835_8b52d6cf1daabf7e0e9be379112846e3.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
2
2018
06
01
Solution to the minimum harmonic index of graphs with given minimum degree
25
33
EN
Meili
Liang
Guangdong University of Foreign Studies
liangmeili2004@163.com
Bo
Cheng
Guangdong University of Foreign Studies
bocheng2006@126.com
Jianxi
Liu
Guangdong University of Foreign Studies
liujianxi2001@gmail.com
10.22108/toc.2017.101076.1462
The harmonic index of a graph $G$ is defined as $ H(G)=sumlimits_{uvin E(G)}frac{2}{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. Let $mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among $mathcal{G}(n,k)$. We solve the problem for each integer $k (1le kle n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$. This result together with our previous result which solve the problem for each integer $k (n/2 le kle n-1)$ give a complete solution of the problem.
harmonic index,minimum degree,extremal graphs
http://toc.ui.ac.ir/article_22272.html
http://toc.ui.ac.ir/article_22272_28d4f6f37d2867d952c1398e234888f8.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
2
2018
06
01
On matrix and lattice ideals of digraphs
35
46
EN
Hamid
Damadi
Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.
hamid.damadi@aut.ac.ir
Farhad
Rahmati
Amirkabir University of Technology
frahmati@aut.ac.ir
10.22108/toc.2017.105701.1510
Let $textit{G}$ be a simple, oriented connected graph with $n$ vertices and $m$ edges. Let $I(textbf{B})$ be the binomial ideal associated to the incidence matrix textbf{B} of the graph $G$. Assume that $I_L$ is the lattice ideal associated to the rows of the matrix $textbf{B}$. Also let $textbf{B}_i$ be a submatrix of $textbf{B}$ after removing the $i$-th row. We introduce a graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I(textbf{B}_i)$ and $I(textbf{B}_i)=I_L$. After that we introduce another graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(textbf{B})=I_L$. It is shown that the heights of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $n-1$ and the dimensions of $I(textbf{B})$ and $I(textbf{B}_i)$ are equal to $m-n+1$; then $I(textbf{B}_i)$ is a complete intersection ideal.
Directed graph,Binomial ideal,Matrix ideals
http://toc.ui.ac.ir/article_22320.html
http://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
7
2
2018
06
01
Reduced zero-divisor graphs of posets
47
54
EN
Deiborlang
Nongsiang
0000-0002-0213-7671
North Eastern Hill University
ndeiborlang@yahoo.in
Promode
Kumar
Saikia
North Eastern Hill University
promode4@gmail.com
10.22108/toc.2018.55164.1417
This paper investigates properties of the reduced zero-divisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a non-annihilator prime ideal. We also classify all posets whose reduced zero-divisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.
poset,reduced zero-divisor graph,annihilator prime ideal
http://toc.ui.ac.ir/article_22311.html
http://toc.ui.ac.ir/article_22311_893dce7cc938e8e23dd5defcadb2c102.pdf